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Singularity and Integrability

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Date and time:Ìý

Tuesday, February 18, 2014 - 5:00pm

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ECOT 226

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The nature of singularities of solutions of equations has long been used as a tool for detecting integrability.Ìý For ordinary differential equations one often looks for equations with solutions that are single-valued about all movable singularities in the complex plane (this is the Painlevé property).Ìý I will present methods for determining when certain classes of equations admit solutions that are algebraic over the meromorphic functions and so have a finite number of branches over any point.

For discrete (or difference) equations, singularity confinement has provided a useful heuristic for identifying integrable equations.Ìý I will show how this intuitive idea can be turned into a rigorous method for calculating various measures of complexity that are known to be good detectors of (non)integrability: algebraic entropy (the growth of degrees of rational iterates), the Nevanlinna characteristic (the growth of complexity of meromorphic solutions) and Diophantine integrability (the growth of heights of solutions over the rational numbers).